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Parametric Representation of Noncommutative Field Theory

Razvan Gurau, Vincent Rivasseau

TL;DR

The paper develops a parametric representation for the renormalizable NC φ^4_4 theory on the Moyal space by introducing hyperbolic polynomials $HU$ and $HV$ that replace the commutative Symanzik polynomials. By converting position constraints into hypermomenta and performing Gaussian integrations, it derives a positive, determinant-based structure for $HU$ and a Pfaffian-based, quadratic-external-argument structure for $HV$, with leading terms controlled by topological Filk moves. This framework yields clear UV power counting, captures genus and broken-face information, and recovers the commutative limit as $s\to0$ (or $\theta\to\infty$ in the chosen conventions). The explicit graph examples illustrate how hyperbolic polynomials encode noncommutative topology and support potential renormalization analyses in parametric space, offering a canonical route beyond matrix base formulations.

Abstract

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable $φ^4_4$ quantum field theory on the Moyal non commutative ${\mathbb R^4}$ space. This representation involves new {\it hyperbolic} polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information.

Parametric Representation of Noncommutative Field Theory

TL;DR

The paper develops a parametric representation for the renormalizable NC φ^4_4 theory on the Moyal space by introducing hyperbolic polynomials and that replace the commutative Symanzik polynomials. By converting position constraints into hypermomenta and performing Gaussian integrations, it derives a positive, determinant-based structure for and a Pfaffian-based, quadratic-external-argument structure for , with leading terms controlled by topological Filk moves. This framework yields clear UV power counting, captures genus and broken-face information, and recovers the commutative limit as (or in the chosen conventions). The explicit graph examples illustrate how hyperbolic polynomials encode noncommutative topology and support potential renormalization analyses in parametric space, offering a canonical route beyond matrix base formulations.

Abstract

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable quantum field theory on the Moyal non commutative space. This representation involves new {\it hyperbolic} polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information.

Paper Structure

This paper contains 10 sections, 6 theorems, 85 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic Polynomials
  3. Notations
  4. Definition of $HU$ and $HV$
  5. Short and Long Variables
  6. The Determinant or First Hyperbolic Polynomial $HU$
  7. Leading terms in the First Polynomial $HU$
  8. The Quadratic form in the External Positions, or Second Hyperbolic Polynomial
  9. Leading terms
  10. Examples

Key Result

Lemma III.1

For any two $n\times n$ matrices $A$ and $B$ let $R=A\otimes I_2+B\otimes\sigma_y$. Then: and:

Figures (10)

  • Figure 1: The First Filk Move on the Sunshine Graph
  • Figure 2: The First Filk Move on the dual of the Sunshine Graph
  • Figure 3: A Rooted Rosette
  • Figure 4: A SuperRosette
  • Figure 5: The Third Filk Move
  • ...and 5 more figures

Theorems & Definitions (6)

  • Lemma III.1
  • Lemma III.2
  • Lemma III.3
  • Lemma III.4
  • Lemma III.5
  • Lemma IV.1